Relations II

CS 441 Discrete Mathematics for CS Lecture 22

Relations II

Milos Hauskrecht [email protected] 5329 Sennott Square

CS 441 Discrete mathematics for CS

M. Hauskrecht

Cartesian product (review) • Let A={a1, a2, ..ak} and B={b1,b2,..bm}. • The Cartesian product A x B is defined by a set of pairs {(a1 b1), (a1, b2), … (a1, bm), …, (ak,bm)}. Example: Let A={a,b,c} and B={1 2 3}. What is AxB? AxB = {(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)}

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Binary relation Definition: Let A and B be sets. A binary relation from A to B is a subset of a Cartesian product A x B. Example: Let A={a,b,c} and B={1,2,3}. • R={(a,1),(b,2),(c,2)} is an example of a relation from A to B.

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Representing binary relations • We can graphically represent a binary relation R as follows: • if a R b then draw an arrow from a to b. ab Example: • Let A = {0, 1, 2}, B = {u,v} and R = { (0,u), (0,v), (1,v), (2,u) } • Note: R  A x B. • Graph: 2 0 1

u v

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Representing binary relations • We can represent a binary relation R by a table showing (marking) the ordered pairs of R. Example: • Let A = {0, 1, 2}, B = {u,v} and R = { (0,u), (0,v), (1,v), (2,u) } • Table: or R | u v R | u v 0 | x x 0 | 1 1 1 | x 1 | 0 1 2 | 1 0 2 | x

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Properties of relations Properties of relations on A: • • • • •

Reflexive Irreflexive Symmetric Anti-symmetric Transitive

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Reflexive relation Reflexive relation • Rdiv ={(a b), if a |b} on A = {1,2,3,4} • Rdiv = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)}

MRdiv

=

1 0 0 0

1 1 0 0

1 0 1 0

1 1 0 1

• A relation R is reflexive if and only if MR has 1 in every position on its main diagonal.

M. Hauskrecht

CS 441 Discrete mathematics for CS

Irreflexive relation Irreflexive relation • R≠ on A={1,2,3,4}, such that a R≠ b if and only if a ≠ b. • R≠={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}

MR

=

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

• A relation R is irreflexive if and only if MR has 0 in every position on its main diagonal.

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Symmetric relation Symmetric relation: • R≠ on A={1,2,3,4}, such that a R≠ b if and only if a ≠ b. • R≠={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}

MR

=

0 1 1 1

1 0 1 1

1 1 0 1

1 1 1 0

• A relation R is symmetric if and only if mij = mji for all i,j.

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Anti-symmetric relation Definition (anti-symmetric relation): A relation on a set A is called anti-symmetric if • [(a,b)  R and (b,a)  R]  a = b where a, b  A. Example 3: • Relation Rfun on A = {1,2,3,4} defined as: • Rfun = {(1,2),(2,2),(3,3)}. • Is Rfun anti-symmetric? • Answer: Yes. It is anti-symmetric

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Anti-symmetric relation Antisymmetric relation • relation Rfun = {(1,2),(2,2),(3,3)}

MRfun =

0 0 0 0

1 1 0 0

0 0 1 0

0 0 0 0

• A relation is antisymmetric if and only if mij = 1  mji = 0 for i≠ j.

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Transitive relation Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • • • • •

Example 1: Rdiv ={(a b), if a |b} on A = {1,2,3,4} Rdiv = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} Is Rdiv transitive? Answer:

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Transitive relation Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • • • • •

Example 1: Rdiv ={(a b), if a |b} on A = {1,2,3,4} Rdiv = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} Is Rdiv transitive? Answer: Yes.

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Transitive relation Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • Example 2: • R≠ on A={1,2,3,4}, such that a R≠ b if and only if a ≠ b. • R≠={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}

• Is R≠ transitive ? • Answer:

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Transitive relation Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • Example 2: • R≠ on A={1,2,3,4}, such that a R≠ b if and only if a ≠ b. • R≠={(1,2),(1,3),(1,4),(2,1),(2,3),(2,4),(3,1),(3,2),(3,4),(4,1),(4,2),(4,3)}

• Is R≠ transitive? • Answer: No. It is not transitive since (1,2)  R and (2,1)  R but (1,1) is not an element of R.

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Transitive relations Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • Example 3: • Relation Rfun on A = {1,2,3,4} defined as: • Rfun = {(1,2),(2,2),(3,3)}. • Is Rfun transitive? • Answer:

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Transitive relations Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • Example 3: • Relation Rfun on A = {1,2,3,4} defined as: • Rfun = {(1,2),(2,2),(3,3)}. • Is Rfun transitive? • Answer: Yes. It is transitive.

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Combining relations Definition: Let A and B be sets. A binary relation from A to B is a subset of a Cartesian product A x B. • Let R  A x B means R is a set of ordered pairs of the form (a,b) where a  A and b  B. Combining Relations • Relations are sets  combinations via set operations • Set operations of: union, intersection, difference and symmetric difference.

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Combining relations Example: • Let A = {1,2,3} and B = {u,v} and • R1 = {(1,u), (2,u), (2,v), (3,u)} • R2 = {(1,v),(3,u),(3,v)} What is: • R1  R2 = ?

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Combining relations Example: • Let A = {1,2,3} and B = {u,v} and • R1 = {(1,u), (2,u), (2,v), (3,u)} • R2 = {(1,v),(3,u),(3,v)} What is: • R1  R2 = {(1,u),(1,v),(2,u),(2,v),(3,u),(3,v)} • R1  R2 = ?

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Combining relations Example: • Let A = {1,2,3} and B = {u,v} and • R1 = {(1,u), (2,u), (2,v), (3,u)} • R2 = {(1,v),(3,u),(3,v)} What is: • R1  R2 = {(1,u),(1,v),(2,u),(2,v),(3,u),(3,v)} • R1  R2 = {(3,u)} • R1 - R2 = ?

CS 441 Discrete mathematics for CS

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Combining relations Example: • Let A = {1,2,3} and B = {u,v} and • R1 = {(1,u), (2,u), (2,v), (3,u)} • R2 = {(1,v),(3,u),(3,v)} What is: • R1  R2 = {(1,u),(1,v),(2,u),(2,v),(3,u),(3,v)} • R1  R2 = {(3,u)} • R1 - R2 = {(1,u),(2,u),(2,v)} • R2 - R1 = ?

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Combining relations Example: • Let A = {1,2,3} and B = {u,v} and • R1 = {(1,u), (2,u), (2,v), (3,u)} • R2 = {(1,v),(3,u),(3,v)} What is: • R1  R2 = {(1,u),(1,v),(2,u),(2,v),(3,u),(3,v)} • R1  R2 = {(3,u)} • R1 - R2 = {(1,u),(2,u),(2,v)} • R2 - R1 = {(1,v),(3,v)}

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Combination of relations Representation of operations on relations: • Question: Can the relation be formed by taking the union or intersection or composition of two relations R1 and R2 be represented in terms of matrix operations? • Answer: Yes

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Combination of relations: implementation Definition. The join, denoted by , of two m-by-n matrices (aij) and (bij) of 0s and 1s is an m-by-n matrix (mij) where • mij = aij  bij for all i,j = pairwise or (disjunction) • • • •

Example: Let A = {1,2,3} and B = {u,v} and R1 = {(1,u), (2,u), (2,v), (3,u)} R2 = {(1,v),(3,u),(3,v)}

• MR1 =1 1 1

0 1 0

MR2 = 0 0 1

1 0 1

M(R1  R2)= 1 1 1

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Combination of relations: implementation Definition. The meet, denoted by  , of two m-by-n matrices (aij) and (bij) of 0s and 1s is an m-by-n matrix (mij) where • mij = aij  bij for all i,j = pairwise and (conjunction) • • • •

Example: Let A = {1,2,3} and B = {u,v} and R1 = {(1,u), (2,u), (2,v), (3,u)} R2 = {(1,v),(3,u),(3,v)}

• MR1 =1 1 1

0 1 0

MR2 = 0 0 1

1 0 1

MR1  MR2= 0 0 1

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Composite of relations Definition: Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of the ordered pairs (a,c) where a  A and c  C, and for which there is a b  B such that (a,b)  R and (b,c)  S. We denote the composite of R and S by S o R. Examples: • Let A = {1,2,3}, B = {0,1,2} and C = {a,b}. • R = {(1,0), (1,2), (3,1),(3,2)} • S = {(0,b),(1,a),(2,b)} • SoR=? CS 441 Discrete mathematics for CS

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Composite of relations Definition: Let R be a relation from a set A to a set B and S a relation from B to a set C. The composite of R and S is the relation consisting of the ordered pairs (a,c) where a  A and c  C, and for which there is a b  B such that (a,b)  R and (b,c)  S. We denote the composite of R and S by S o R. Examples: • Let A = {1,2,3}, B = {0,1,2} and C = {a,b}. • R = {(1,0), (1,2), (3,1),(3,2)} • S = {(0,b),(1,a),(2,b)} • S o R = {(1,b),(3,a),(3,b)} CS 441 Discrete mathematics for CS

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Implementation of composite Definition. The Boolean product, denoted by , of an m-by-n matrix (aij) and n-by-p matrix (bjk) of 0s and 1s is an m-by-p matrix (mik) where 1, if aij = 1 and bjk = 1 for some k=1,2,...,n • mik = 0, otherwise Examples: • Let A = {1,2,3}, B = {0,1,2} and C = {a,b}. • R = {(1,0), (1,2), (3,1),(3,2)} • S = {(0,b),(1,a),(2,b)} • S o R = {(1,b),(3,a),(3,b)}

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Implementation of composite Examples: • Let A = {1,2}, B= {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 1 1 1 0 0 0 MS = 0 0 MR = 1 1 1 MR  MS = ?

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Implementation of composite Examples: • Let A = {1,2}, {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 1 1 1 0 MR = 1 0 0 MS = 0 0 1 1 = x x MR  MS x x

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Implementation of composite Examples: • Let A = {1,2}, {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 1 1 1 0 0 0 MS = 0 0 MR = 1 1 1 MR  MS = 1 x x x

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Implementation of composite Examples: • Let A = {1,2}, {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 MR = 1

1 0

1 0

MS

MR  MS

=

1 x

1 x

=

1 0 1

0 0 1

M. Hauskrecht

CS 441 Discrete mathematics for CS

Implementation of composite Examples: • Let A = {1,2}, {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 MR = 1

1 0

1 0

MS

MR  MS

=

1 1

1 x

=

CS 441 Discrete mathematics for CS

1 0 1

0 0 1

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Implementation of composite Examples: • Let A = {1,2}, {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 1 1 1 0 MR = 1 0 0 MS = 0 0 1 1 = 1 1 MR  MS 1 0 MS  R =

? CS 441 Discrete mathematics for CS

M. Hauskrecht

Implementation of composite Examples: • Let A = {1,2}, {1,2,3} C = {a,b} • R = {(1,2),(1,3),(2,1)} is a relation from A to B • S = {(1,a),(3,b),(3,a)} is a relation from B to C. • S  R = {(1,b),(1,a),(2,a)} 0 1 1 1 0 0 0 MS = 0 0 MR = 1 1 1 MR  MS = 1 1 1 0 MS  R =

1 1

1 0

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Composite of relations Definition: Let R be a relation on a set A. The powers Rn, n = 1,2,3,... is defined inductively by • R1 = R and Rn+1 = Rn  R. Examples • R = {(1,2),(2,3),(2,4), (3,3)} is a relation on A = {1,2,3,4}. • R1= ?

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Composite of relations Definition: Let R be a relation on a set A. The powers Rn, n = 1,2,3,... is defined inductively by • R1 = R and Rn+1 = Rn  R. Examples • R = {(1,2),(2,3),(2,4), (3,3)} is a relation on A = {1,2,3,4}. • R 1 = R = {(1,2),(2,3),(2,4), (3,3)} • R2=?

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Composite of relations Definition: Let R be a relation on a set A. The powers Rn, n = 1,2,3,... is defined inductively by • R1 = R and Rn+1 = Rn  R. Examples • R = {(1,2),(2,3),(2,4), (3,3)} is a relation on A = {1,2,3,4}. • R 1 = R = {(1,2),(2,3),(2,4), (3,3)} • R 2 = {(1,3), (1,4), (2,3), (3,3)} • R3=?

CS 441 Discrete mathematics for CS

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Composite of relations Definition: Let R be a relation on a set A. The powers Rn, n = 1,2,3,... is defined inductively by • R1 = R and Rn+1 = Rn  R. Examples • R = {(1,2),(2,3),(2,4), (3,3)} is a relation on A = {1,2,3,4}. • R 1 = R = {(1,2),(2,3),(2,4), (3,3)} • R 2 = {(1,3), (1,4), (2,3), (3,3)} • R 3 = {(1,3), (2,3), (3,3)} • R4=?

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Composite of relations Definition: Let R be a relation on a set A. The powers Rn, n = 1,2,3,... is defined inductively by • R1 = R and Rn+1 = Rn  R. Examples • R = {(1,2),(2,3),(2,4), (3,3)} is a relation on A = {1,2,3,4}. • R 1 = R = {(1,2),(2,3),(2,4), (3,3)} • R 2 = {(1,3), (1,4), (2,3), (3,3)} • R 3 = {(1,3), (2,3), (3,3)} • R 4 = {(1,3), (2,3), (3,3)} • R k = ? , k > 3.

CS 441 Discrete mathematics for CS

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Composite of relations Definition: Let R be a relation on a set A. The powers Rn, n = 1,2,3,... is defined inductively by • R1 = R and Rn+1 = Rn  R. Examples • R = {(1,2),(2,3),(2,4), (3,3)} is a relation on A = {1,2,3,4}. • R 1 = R = {(1,2),(2,3),(2,4), (3,3)} • R 2 = {(1,3), (1,4), (2,3), (3,3)} • R 3 = {(1,3), (2,3), (3,3)} • R 4 = {(1,3), (2,3), (3,3)} • R k = R 3, k > 3.

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Transitive relation Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • • • • •

Example 1: Rdiv ={(a b), if a |b} on A = {1,2,3,4} Rdiv = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} Is Rdiv transitive? Answer: ?

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Transitive relation Definition (transitive relation): A relation R on a set A is called transitive if • [(a,b)  R and (b,c)  R]  (a,c)  R for all a, b, c  A. • • • • •

Example 1: Rdiv ={(a b), if a |b} on A = {1,2,3,4} Rdiv = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,4), (3,3), (4,4)} Is Rdiv transitive? Answer: Yes.

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Connection to Rn Theorem: The relation R on a set A is transitive if and only if Rn  R for n = 1,2,3,... . Proof: biconditional (if and only if) (  ) Suppose Rn  R, for n =1,2,3,... . • Let (a,b)  R and (b,c)  R • by the definition of R  R, (a,c)  R  R =R2  R • Therefore R is transitive.

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Connection to Rn Theorem: The relation R on a set A is transitive if and only if Rn  R for n = 1,2,3,... . Proof: biconditional (if and only if) () Suppose R is transitive. Show Rn  R, for n =1,2,3,... . • Let P(n) : Rn  R. Mathematical induction. • Basis Step:

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Connection to Rn Theorem: The relation R on a set A is transitive if and only if Rn  R for n = 1,2,3,... . Proof: biconditional (if and only if) () Suppose R is transitive. Show Rn  R, for n =1,2,3,... . • Let P(n) : Rn  R. Mathematical induction. • Basis Step: P(1) says R1 = R so, R1  R is true.

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Connection to Rn Theorem: The relation R on a set A is transitive if and only if Rn  R for n = 1,2,3,... . Proof: biconditional (if and only if) () Suppose R is transitive. Show Rn  R, for n =1,2,3,... . • Let P(n) : Rn  R. Mathematical induction. • Basis Step: P(1) says R1 = R so, R1  R is true. • Inductive Step: show P(n)  P(n+1) • Want to show if Rn  R then Rn+1  R.

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Connection to Rn Theorem: The relation R on a set A is transitive if and only if Rn  R for n = 1,2,3,... . Proof: biconditional (if and only if) () Suppose R is transitive. Show Rn  R, for n =1,2,3,... . • Let P(n) : Rn  R. Mathematical induction. • Basis Step: P(1) says R1 = R so, R1  R is true. • Inductive Step: show P(n)  P(n+1) • Want to show if Rn  R then Rn+1  R. • Let (a,b)  Rn+1 then by the definition of Rn+1 = Rn  R there is an element x  A so that (a,x)  R and (x,b)  Rn  R (inductive hypothesis). In addition to (a,x)  R and (x,b)  R, R is transitive; so (a,b)  R. • Therefore, Rn+1  R. CS 441 Discrete mathematics for CS

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Number of reflexive relations Theorem: The number of reflexive relations on a set A, where | A | = n is: 2 n(n-1) . Proof: • A reflexive relation R on A must contain all pairs (a,a) where a  A. • All other pairs in R are of the form (a,b), a ≠ b, such that a, b  A. • How many of these pairs are there? Answer: n(n-1). • How many subsets on n(n-1) elements are there? • Answer: 2 n(n-1) .

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